# Model categories of quiver representations

## TL;DR

This paper extends Gillespie's theorem to construct model category structures on categories of quiver representations, including generalized chain complexes and periodic complexes, broadening the scope of homological algebra tools.

## Contribution

It generalizes Gillespie's theorem to a wider class of self-injective quivers with relations, enabling systematic construction of model structures on various representation categories.

## Key findings

- Constructs model category structures on categories of quiver representations.
- Includes categories of N-periodic and N-complexes as special cases.
- Provides a unified framework for generalized chain complexes.

## Abstract

Gillespie's Theorem gives a systematic way to construct model category structures on $\mathscr{C}( \mathscr{M} )$, the category of chain complexes over an abelian category $\mathscr{M}$. We can view $\mathscr{C}( \mathscr{M} )$ as the category of representations of the quiver $\cdots \rightarrow 2 \rightarrow 1 \rightarrow 0 \rightarrow -1 \rightarrow -2 \rightarrow \cdots$ with the relations that two consecutive arrows compose to $0$. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of $N$-periodic chain complexes, the category of $N$-complexes where $\partial^N = 0$, and the category of representations of the repetitive quiver $\mathbb{Z} A_n$ with mesh relations.